[jyl]

Suppose the edge of the large coin were cut off, and straightened into a line segment. It has length 2 * pi * 3. The circumference of the small coin is 2 * pi * 1. So the small coin needs to roll 3 times to get from one end of the line segment to the other. Exactly as you said.

But - the edge of the large coin is not a line segment. It is a circular arc that makes a full 360 degree loop. Thus the small coin makes 1 full rotation just to follow this arc. Even if the small coin were slid around the large coin's edge (the same point on the small coin's edge is always touching the large coin), it would make 1 full rotation by the time it returned to its original location.

3 + 1 = 4. The small coin makes 4 full rotations around its own center.

Test this. Take 2 quarters. Place them so that the 6 o'clock of the rotating quarter touches the 12 o'clock of the fixed (bottom) quarter. Orient the rotating quarter so that the top of Washington's head points at 12 o'clock. Now roll the rotating quarter around the fixed quarter until it is back at its starting point. How many times does the top of Washington's head point at 12 o'clock? Once, or twice?

Now, imagine the small coin were rotated around the "inside" of the large coin's edge. What's the answer then?

I'm told the professor who wrote this SAT question got the wrong answer himself.

I think the thing here is, our intuition is familiar with round things rolling on straight things (wheels on a road). We are not accustomed to thinking of round things rolling on other round things. Unless you're an engineer designing planetary gears.